Exercise 11, § 5

Let $p$ be a prime number. Then $p$ is said to be a torsion prime number of G if there exists a connected closed subgroup H of G, of maximum rank, such that the $p$-torsion subgroup of $\pi_1(H)$ is non-zero. Put $R = R(G,T)$.

a) The torsion prime numbers of G are the torsion prime numbers of R (Exerc. 10) and the prime numbers that divide the order of the torsion group of $\pi_1(G)$.

b) Show that every torsion prime number of G divides $w/l$!, where $w$ is the order of W and $l$ the rank of the semi-simple group D(G) (use Chap. VI, §2, no. 4, Prop. 7).

c) Let $t\in T$ and $n\in \mathbf{N}$ be such that $t^n\in C(G)$. Let $R_1$ be the set of $\alpha \in R$ such that $\alpha (t) = 1$, and $m$ the order of the torsion group of $Q(R^{\vee})/Q(R^{\vee}_1)$ (cf. Exerc. 10). Show that every prime factor of $m$ divides $n$ (reduce to the case in which G is simply-connected and almost simple). d) Assume that G is simply-connected. Let $g\in G,n\in \mathbf{N}$ be such that $g^n$ belongs to C(G) and such that no torsion prime number of G divides $n$. Show that the derived group of the centralizer of $g$ is simply-connected.

e) Let $g$ be an element of G and $p$ a prime number, such that $g^p=e$ and $p$ is not a torsion prime number of G. Show that the centralizer $Z(g)$ is connected and that $p$ is not a torsion prime number of $Z(g)$.

f) Assume that G is simply-connected, and let $p$ be a torsion prime number of G. Prove that there exists an element $g$ of G of order $p$ such that $\pi_1(Z(g))$ is cyclic of order $p$ (use Exerc. 10 and Exerc. 4 of Chap. VI, §4).